Does anyone have an idea? Close friends? I usually see this term in stories where the friends were best friends and then they fight for some reason.
2 : firmly loyal <became fast friends>
Of people: steadfast, with unwavering feeling. (Now only in set phrases like "fast friend".) [from 10th c.]
Etymonline provides some background:
O.E. fæst "firmly fixed, steadfast, secure, enclosed," probably from P.Gmc. *fastuz (cf. O.Fris. fest, O.N. fastr, Du. vast, Ger. fest)
The sense of "fast" here is almost obsolete; it's the "fast" of "steadfast," as noted by RegDwight above. It means "tight" or "secure," and you'll see it in literature in expressions like "hold fast" meaning to "get a tight grip" (on something). "Hold fast to your dreams" means to keep them close and never let them go.
Which is exactly what you should do with a good friend.