Learn French: Je suis un homme.

In French, all nouns have a gender (masculine or feminine), and the word for 'a' or 'an' changes to match it. Because homme (man) is a masculine noun, we use the masculine article un. If it were a feminine noun like femme (woman), you would use une.

When I listen to this, it sounds like there is an 'n' sound connecting the words. Why is that?

This is due to a pronunciation rule called liaison. The word un normally has a nasal sound with a silent 'n' at the end. However, because homme starts with a silent 'h', it acts like a word starting with a vowel. The silent 'n' wakes up and connects to the next word, making it sound like 'un-nomme'.

Is the 'h' at the beginning of homme ever pronounced?

No, the 'h' in homme is completely silent, just like in the English word 'hour' or 'honor'. You pronounce the word starting right from the 'o' sound.

I thought the verb 'to be' was être. Why does it say suis?

suis means 'am'. It is the conjugated form of the verb être (to be) that specifically goes with Je (I). Just like in English where we change 'be' to 'am' when we say 'I am', in French être changes to suis for Je.