Learn Turkish: Ben iyiyim.

The word iyi (good or fine) ends in a vowel, and the "I am" suffix -im starts with a vowel. In Turkish, you usually cannot put two vowels right next to each other. The y acts as a "buffer" letter to separate them, making it iyi-y-im.

Why does the ending use an "i" (-im) instead of another vowel?

This is because of 4-way vowel harmony. The suffix changes to match the last vowel of the root word. Since the last vowel in iyi is an i, the suffix also uses an i, becoming -im. If the word had ended in an a or ı, the suffix would have been -ım.

Since the ending means "I am," do I actually need to say Ben?

No, you can simply say İyiyim. The -im ending already tells the listener that it is "I" who is fine. Using the pronoun Ben is optional and is usually just done for emphasis, like saying "As for me, I am fine."