Learn Russian: Оба брата значительно постарели за эти два года. - Both brothers have aged significantly over these two years.

The lesson focuses on на + amount + comparative adjective (like на два года старше). Here, the comparative idea is built into the verb постарели (to become older). Because значительно (significantly) is an adverb directly modifying the verb, it doesn't need на. If you used the comparative adjective старше, you would use на, like стали старше на много лет.

Does the prefix по- in постарели mean they aged 'a little bit'?

Not in this case. While по- often means doing something for a short time or a little bit (like поработать), when added to verbs derived from adjectives, it simply forms the perfective aspect to show a completed change of state. Постареть just means 'to age' or 'to grow old'. Combined with значительно, they clearly aged a lot!

Why do we use за + Accusative (за эти два года) instead of в течение + Genitive for 'over these two years'?

Both can translate to 'during' or 'over,' but за + Accusative is used to show the time it took to reach a completed result. Because the sentence uses the perfective verb постарели to emphasize a finished change in their appearance, за is the natural choice to mark the timeframe of that completed change.