Looking at the Semag, Aseh 26, it says that there are two posukim that obligate the mitzvah of tzitzit. One is BaMidbar 15:38 and the other is Devarim 22:12 which says:
גדילים תעשה לך על ארבע כנפות כסותך אשר תכסה בה
This means that there are two parts to this mitzvah. The first part is the placing of the g'dilim, meaning the four threads which are doubled over to make eight strings, on the four corners of the garment (גדילים תעשה לך על ארבע כנפות כסותך). And the second part of the mitzvah is to be covered, meaning enwrapped, by them (אשר תכסה בה).
This follows the statement found in the Kol Bo, Din Hilchot Tzitzit which says:
והעושה ציצית לעצמו מברך אקב״ו על מצות ציצית ושהחיינו. והרמבם ז״ל כתב שאין מברכים על הציצית בשעת עשיה לפי שאין המצוה אלא להתעטף בו כו׳ וה״ר יצחק כתב בטלית קטן יברך על מצות ציצית ובטלית של עטוף להתעטף.
One who makes tzitzit for himself makes the blessing, "who has sanctified us through His commandments and commanded us concerning the commandment of tzitzit" and the blessing of Shehechiyanu. And the Rambam z"l wrote that we don't make a blessing on the tzitzit at the time of making it because there is no commandment except to be wrapped in it. Etc. And our teacher, Rabbi Yitzchok wrote in regard to the tallit katan, one will bless, "al Mitzvah tzitzit and on a tallit capable of wrapping (של עטוף) one will bless l'hitatef.
This idea is explained further in HaIttur, Chelek 2, Sha'ar HaTzitzit, Sha'ar 3: Sha'ar Adam, Chelek 2, which says that this is a disagreement among the Sages as to whether there is a mitzvah concerning the garment or only in regard to the wrapping.
The first opinion is that if you have a four cornered garment which is large enough to require tzitzit, you are obligated to make tzitzit on it whether you are using it at that moment or not. And this is the source for the text of the blessing, "al mitzvat tzitzit. This first opinion says that the blessing of l'hitatef is said at the actual time of being enwrapped in the garment, but if you didn't need that particular garment when you made the tzitzit, you don't say l'hitatef.
The second opinion is that you only make a blessing when the Mitzvah is completed and that is specifically when you wrap in the garment which has tzitzit on it.
The Ittur points out that his tradition follows the view of Rav Chisda, that any mitzvah where making something is the completion of the act, you are required to make the blessing. And any mitzvah where making something does not complete the act, meaning something else is done in regard to it, does not require a blessing in regard to making it.
The Ittur points out that this view maintains both opinions, that there is a Mitzvah in regard to the garment itself even though no blessing is required to be said at the time the tzitzit are placed on the garment and in regard to the second view that the blessing must be said when the person wraps in the tallit.
It is worth noting from the Kol Bo that with a tallit that is too small for atifah one falls back to the blessing of making the tzitzit on ones four cornered garment.
The Semag goes on to explain from Menachot 40b which says in the name of Shmuel:
תכלת אין בה משום כלאים ואפי' בטלית פטורה
מאי טלית פטורה
אילימא דלית בה שיעורא והתניא
טלית שהקטן מתכסה בו ראשו ורובו
that the minimum measure of the four cornered garment that is obligated to tzitzit is:
טלית שהקטן מתכסה בה ראשו ורובו והגדול יוצא בה עראי חייבת בציצית, אין הקטן תכסה בה ראשו ורובו אף על פי שהגדול יוצא בה עראי פטורה
It must be large enough to cover the head and the majority of the body of the child. (טלית שהקטן מתכסה בה ראשו ורובו...חייבת בציצית)
It also distinguishes here between the child and the adult. With the child, if it is large enough to cover him completely, then the tallit requires tzitzit and the child fulfills completely by making the blessing of l'hitatef b'tzitzit.
That same size tallit, which is large enough for the child to fulfill the mitzvah completely is not large enough for an adult to fulfill completely. But in a temporary situation, (in the language of some sources it is called בשעת דחק, in others אקראי) the adult can rely upon it to fulfill in a minimal way (והגדול יוצא בה עראי). The adult cannot do atifah, which is the complete way of fulfilling, but he can have tzitzit on a four cornered garment.
If the garment doesn't have the size required for the child to fulfill properly, then the garment is not required to have tzitzit at all. This would be considered like a scarf.
The Semag goes on to emphasize that the size for covering sufficiently is dependent upon the size of the person wearing the garment. That means the size required for the katan, the child, is smaller than the size required for the adult.
The Semag clarifies more about the katan and what his physical size will be from Sukkah 42a:
קטן שיודע להתעטף בציצית חייב בציצית.
That it's not any size child, meaning for example, like a child of five or six, but one who knows how to be enwrapped. It also introduces the word atifah (עטיפה) here as a synonym for being covered (תכסה) around the head and the majority of the torso. This also follows the language of the Ittur cited above, Sha'ar 2: Sha'ar Tallit, Chellek 3 which says:
אע״ג דמברכינין להתעטף בציצית לשון כסות היא כדכתיב אשר תכסה בה
And this distinction of Atifah (עטיפה) is based upon Moed Katan 24a which says in the name of Shmuel:
וכל עטיפה שאינה כעטיפת ישמעאלים אינה עטיפה
That the wrapping practiced by the Yishmaelim is around the head with the face showing and around the shoulders down to the chest as described by the Alter Rebbe in Siddur Tehillat HaShem, page 11 in conjunction with the blessings for tzitzit.
This same reference to Moed Katan is also said in the name of the Gaonim in Tur, Orach Chaim, Hilchot Tzitzit, Siman 8.
This same expression is brought in Sefer Orchot Chaim, Hilchot Tzitzit, sief 21 which clarifies further:
ושמעתי שהקטן רוצה לומר קטן שהגיע לחינוך כדאמרינן קטן היודע להתעטף חייב בציצית
This means that the katan is Bar Mitzvah age like is understood from Avot 5:22. That also has implications about the minimum size of the tallit katan and when a garment is required to have tzitzit. Something that could cover a younger child isn't really required to have tzitzit at all.
The Sefer Orchot Chaim goes on to say in sief 22:
והעושה ציצית לעצמו מברך על מצות ציצית ושהחיינו והרמב״ם ז״ל כתב שאין מברכים על הציצית בשעת עשייה לפי שאין מצוה אלא להתעטף בו.
Two ideas are brought out here. The first is that the blessing of al mitzvah tzitzit pertains to the placing of tzitzit on a garment that requires them. And the second idea according to Rambam is that the primary aspect of the mitzvah and its completion is the being wrapped in them. This is like is understood from the posuk cited above from Devarim 22:12.
The Sefer Orchot Chaim goes further in sief 23 to say:
המתעטף בטלית מצויצת מברך להתעטף בציצית והרב יהודה חסיד כתב בטלית קטן יברך על מצות ובטלית של עטוף להתעטף. ואם לא שח בין טלית קטן לתליט גדול כתב רבינו מאיר נ״ע שאין צריך לברך לגדול.
So it appears from Sukkah 42a that atifah is a synonym for being completely covered and concealed by the garment which is defined as covering your head and the majority of your body.
And this would be the basis for the language of the blessing to be wrapped (or completely covered) by the tzitzit.