Why does an increase in GDP lead to an increase average incomes? I am a high-school student...sorry if this is a very simple question.

Is it due to the fact that, as there is more money in the economy, firms can afford to pay higher wages/employ more people? But surely this effect is more or less cancelled out by the increases in average price level within an economy?

Thank you in advance.
In a nutshell, a higher GDP is the very same as an increase in average incomes, at least if the population remains constant. If there is more stuff produced and consumed, overall income is higher and average income as well.

As you rightly point out, since income is measured in monetary terms, one has to correct for price increases. And this is usually done. Whenever you hear 'GDP has increased', this better be in real terms, i.e. in constant prices. In nominal terms, GDP (as well as tax revenues) are increasing almost every year, which renders nominal comparisons usually meaningless.