I think what this is saying is that one should treat the words properly.
There are two ingredients to a proper tefilla as Hashem needs it. The first is with kavana, which means putting one's soul into it; concentrating and contemplating the words, feeling the meaning of the words with love and fear; meaning it sincerely. If you would like further information about davening with kavana in english, I recommend purchasing a "Pathway to Prayer" book and reading the introductions*.
The other is davening with dveykut, with cleaving. This means even if you don't know what you are saying (and what business is it of ours anyway), you do know that these are not your words, they are Hashem's precious words. They are profound, powerful words written by prophets, that fulfil the needs of Above.
ולא לחינם אמרו החכמים דתפילה צורך גבוה הוא
It wasn't for nothing that the Sages said that prayer is the needs of Above
(Aruch Hashulchan OC 89:8)
Saying them with dveykut means treating them with awe, saying them slowly and carefully like handling fine china. It is intimately connected, and aimed at bringing about a cleaving to Hashem, and it therefore means being aware of Who you are saying them to, and how much you mean to Him, and how important your tefilla is for Him, how precious these words are to Him, and being aware that this is in His service, not our own. Indeed, one can't cleave to someone else if one is busy with one's own needs.
The mashal is told of the wife who complains to her husband that he doesn't say "I love you" enough. He retorts that she knows he loves her, he will tell her if that changes, and therefore he doesn't need to say it. She doesn't accept this: she needs to hear it, it's not about his need. He moans and says "if I say it, will it make you happy?" and when she affirms, he grumpily (with a smirk) says "fine, I love you, now will you let me read my newspaper?". She is indeed happy, even though she would agree he still needs to keep working on his kavana.
Here is a short shiur that discusses this, and you can find several others on this channel. Rabbi Friedman is a great teacher of Chabad Chassidus, whose Rebbeim have made it their mission to bring down the teachings of the Besht to every day life, and this answer is an introduction to how they teach their chassidim to daven, and something that has improved my tefilla service greatly upon learning (I am not a Chabad chassid).
* Note, I would say that the appendix on the word "Baruch" would be hard to fit with the teaching of the Besht you are bringing without a fuller explanation, and therefore please bear in mind that there are opinions that either oppose or clarify what is explained there.
