Learn Hardcore Serbian: Rekla sam zbogom prijateljici.

In standard Serbian, the auxiliary sam is a clitic. Clitics usually go in the second position of the sentence or clause, not at the very beginning.

So:

Rekla sam zbogom prijateljici. = natural
Ja sam rekla zbogom prijateljici. = also natural, because sam is now in second position after ja
Sam rekla... = not standard

This is a very common pattern in Serbian past tense.

What does rekla mean, and why does it end in -la?

Rekla is the past active participle of the verb reći (to say).

The ending shows the gender and number of the speaker:

rekao = a male speaker said
rekla = a female speaker said
rekli = plural, or mixed/male group said
re kle = feminine plural said

So Rekla sam... tells you the speaker is female.

What is sam doing in the sentence?

Sam is the 1st person singular present form of biti (to be), and Serbian uses it to form the past tense.