Learn French: Ma voiture est bleue.

You can replace the noun phrase Ma voiture with the possessive pronoun La mienne (mine). Because voiture is a feminine noun, you must use the feminine pronoun to get La mienne est bleue.

I am a male speaker. Should I say 'mon voiture' instead of 'ma voiture'?

No, you still say ma voiture. In French, possessive words agree with the thing being owned, not the person who owns it. Since voiture is feminine, it takes the feminine word ma regardless of who is speaking.

How would I say 'His car is blue' versus 'Her car is blue'?

In French, they are exactly the same: Sa voiture est bleue. Just like ma, the word sa agrees with the feminine car, not the owner. You would only know if it belongs to a man or a woman from the surrounding context.

Why does the word bleue have an 'e' at the end here?

French adjectives have to match the gender of the noun they describe. Because voiture is a feminine noun, you must add an 'e' to the base masculine color bleu to make it feminine.