This midrash does not exist in Bereishit Rabba. Indeed, the verse referenced by the translation "Behold the man whose name is Zemach [branch], and he shall branch up out of his place" is Zechariah 6:12 (see here), and that verse does not appear in Bereishit Rabbah 35 or 37 — or in fact in Bereishit Rabba at all — as can be seen by searching this page (after letting it load) for words of the verse (even צמח will do). Searching Sefaria (which has a good number of classic midrashim) for "צמח שמו" finds only 12 hits in the Midrash and Talmud categories (see here), and none appear to be your midrash.
Instead, this midrash is allegedly found in Bereishit Rabbati, which is a medieval midrash attributed to R. Moshe HaDarshan. Such an attribution (for the ch 37 midrash) can be seen in Abarbanel's Yeshu'ot M'shiḥo (p. 217 of PDF), but the midrash isn't found in the Bereishit Rabbati that we have today, and neither is the one from ch 35 (even Abarbanel only says that the midrash is "said to be from Bereishit Rabbati"). The citations that you are seeing are probably from Pugio Fidei, a medieval Christian polemic by Raymond Martini that allegedly contains extracts of R. Moshe HaDarshan's midrash. The accuracy of these citations is in dispute (as you can see on the linked page).
The midrash Bereishit Rabbati is not well-known, in my experience, and hardly forms a "standard interpretation" of Jewish beliefs (besides the fact that many midrashim are contradictory, and one can't take all midrash at face-value; see especially the question Is belief in midrashim optional?). Add to that the midrash's late authorship (where the Jewish author was certainly aware of Christianity and its claims) and the fact that the midrash may or may not be in the original, and it's likely that some misinterpretation (to be generous) is going on here.
But as you asked for an interpretation, let's give the full text of the midrash (the ch 37 one) that Abarbanel cites.
ויאמר אליהם ראובן אל תשפכו דם (בראשית לז, כב), זה שאמר הכתוב: יתומים
היינו ואין אב (איכה ה, ג). אמר רבי ברכיה: אמר הקב״ה לישראל אתם אמרתם
לפני: יתומים היינו ואין אב, אף גואל שאני עתיד להעמיד מכם - אין לו אב,
שנאמר: הנה איש צמח שמו ומתחתיו יצמח (זכריה ו, יב). וכן הוא אומר: ויעל
כיונק לפניו (ישעיה נג, ב). ועליו אמר דוד: אני היום ילדתיך (תהלים ב, ז).
And Reuben went on, "Shed no blood! ... " (Genesis 37:22), this is like what the verse says: "We have become orphans, fatherless" (Lamentations 5:3). R. Berachya says: the Holy One said to Israel, "you said to me 'we have become orphans, fatherless', so too the redeemer that I will bring from among you will be fatherless, as it says: 'a man called the Branch shall branch out from the place where he is' (Zechariah 6:12). And it also says: 'for he has grown, by His favor' (Isaiah 53:2). And David said about him: 'I have fathered you this day' (Psalms 2:7)."
Abarbanel says that "father" here means "teacher" — the Messiah will learn everything via prophecy — and in fact bring back prophecy to Israel — and not gain his wisdom from a human teacher. He then points out that part of the midrash (including the Lamentations verse) is similar to what's found in Eicha Rabba, where the redeemer refers to Queen Esther, and "no father" means "orphan". Abarbanel then cites a midrash on Isaiah 53:2 that we no longer have, saying that when the Messiah enters the world, his father will die. He then gives another understanding of "no father" as "no comforter", and also points out that if God is bringing the Messiah from "among you", the Messiah must be like us. He has a few more comments, but the gist of it is here.
Please be wary of citations to Jewish sources from Christian sites, especially from missionary sources.
Props to the other answerers (Yaacov Deane, Derdeer) for locating the citation.