The first part, multiplier = 1; initializes the counter variable starting value.
The second part multiplier <= 10; is the logic expression designed to express the end of the count. This is important to distinguish this to the logic expression found in the while statement. Logic expression in while statement may not be considering the end of a count. For example a while statement may consider logic of true or false of reaching the end of a file of unknown length.
The last part multiplier ++ is an expression that changes the counter variable in such a way the loop will eventually produce a false value for the logic expression.
This is a versatile part of the for statement. You can increment or decrement the counter variable in any conceivable mathematical formula.
Consider multiplier -- for counting backwards by 1. Consider multiplier += 2 for counting by 2.
Consider multiplier -= 2 for counting backwards by 2.
Consider multiplier += multiplier^2 for counting by the current multiplier to the power of 2 or by 2, 8, and 20 in the above example. Just remember this last part must always be about changing the counter variable.
Also the counter variable may be changed inside the for loop. Be very careful of doing this because you now have two formulas impacting the counter and you can get confused. I point this out because this is done and sometimes a possible programming solution.
For example the need to start the loop over from inside the loop may have a selection structure detecting this need and changing the counter variable to a new value or the same starting value. Advice is to resist or consider a while loop instead.