Hi there. Happy to!
When we think of buying stocks, we're primarily doing so for a chunk of the company's earnings. So, in a very simple form to show you the benefits of a share buyback, lets assume a few things:
The company's market capitalization: $1 billion
The company's total shares outstanding: 1 billion
The company's earnings: $100 million
So in this situation, considering share price is simply the market cap divided by its total shares outstanding, we get a share price of $1 ($1B market cap divided by 1B in shares outstanding).
From there, we can see that the company has earned $0.10 per share ($100M in profits divided by 1 billion in shares outstanding). With a $1 share price, this means the company is trading at 10 times its earnings.
Lets say the company wants to buy back 10% of its shares outstanding, and it does so. We now have $100M in profits, but we only have 900M shares outstanding instead of 1B. So instead of earning the ten cents per share you were prior to the buyback, you're now earning $0.111 per share ($100M in profits divided by 900M in shares outstanding).
Now that we have $0.111 in earnings per share, if the market were to value the company at the same price to earnings ratio (10x) we now have a share price of $1.11.
Does this make sense? This is a very simple explanation and share buybacks are FAR from guaranteed to raise the share price. But, this is ultimately how it would benefit shareholders.