Mae is the verb. Welsh is in general - Verb/Subject/Object in structure. In this sentence we have 'Mae' - 'is' 'yr aderyn' - 'the bird' 'eisiau gweld' - 'wanting seeing' 'pryf' - 'insect'. Putting that together we have the word for word translation. 'Is the bird wanting seeing insect'. Since there's no question mark we can rearrange the word order in English. 'The bird is wanting seeing insect' which is analogous to the original sentence.

you are right it is " yr aderyn" - but 'yr' itself shortens further to 'r after a word ending in a vowel 'Mae'r aderyn yn - The bird is' 'mae' ends with a vowel (e)