What we are incrementing with a value of one.
You are increasing the parameter run which is a public/global and special variable name for LTspice's internal use.
Is this something like in the 1st iteration we take resistor value as R = R+Tolerance and in 2nd it is like R-Tolerance, something like that?
No. mc() function is something like a random generator between the given boundaries with a uniform distribution. In your circuit, each resistor value will be a random between their minimum possible and maximum possible values (i.e. including tolerance). And this function takes the parameter run (as stated before, a special variable name for LTspice, just like time) as input, so you need to manually increase it at reach iteration. Or else the random generator will generate the same number at each iteration.