be-
"be- makes a verb transitive"
This is the general rule. The examle "greifen" can show what is meant. It is a complicated, but typical case.
? Ich greife einen Stein.
Duden has this first example, under the first meaning which is "1. ergreifen". I did not even know you can use it transitively...but it is rare. Duden's next example is reflexive "sich greifen." All these uses are more common with pure "greifen":
Ich greife mir einen Stein. (to get equipped with)
Ich greife nach einem Stein. (to reach for)
Ich ergreife einen Stein. (to pick up)
"greifen nach etw." is not the same as "etw. ergreifen", both grammatically and semantically.
While the 3 modifications above keep the original meaning, the be- is used to make a semantic shift - from things to ideas, from hand to mind.
Now you can say:
Das begreife ich.
Das habe ich begriffen.
Transitive (but not with hands!), and it does not even matter if present or past tense.
The constant - what you get for free - is the "-griffen", the grammtical form.
Similar patterns ban be formed with:
kommen - bekommen (Der Sommer kommt / Wir bekommen Sommer...i.e. if something comes to you, you recieve it. See re-capere. Also be-come.)
gehen - begehen (begehen = "officially" walk (on) or celebrate, or commit)
halten - behalten
nehmen - sich benehmen (s. benehmen = to be-have)
rühren - berühren (move/stir vs. touch)
stehen - bestehen (stand vs. exist)
be- can also turn nouns and adjectives into verbs:
schön - etw. beschönigen (to pretty up sth.)
Gnade - jmdn. begnadigen
So why should an English speaker say:
Com-pris in French means under-stood, I com-prehend. But what does
ver-standen mean?