Because of the rule that says: with nouns ending in unstressed -el or -er, the "e" of the definite ending (-en or -et) is dropped. Thus: en fågel (a bird) : fågeln (the bird) en fjäder (a feather) : fjädern (the feather) ett exempel (an example) : exemplet (the example) ett finger (a finger) : fingret (the finger)
I'm not a native speaker, so take this with a grain of salt:
From what you understand (fågel and dricker), the sentence translates either to "a bird is drinking" or "the bird is drinking". Since you cannot omit the indefinite article (en/ett), "a bird is drinking" would be "en fågel dricker". Since this is very different from what you hear, it can only be "the bird is drinking", which translates to "fågeln dricker".
Apart from that, your listening comprehension will improve over time.