What we are incrementing with a value of one.
You are increasing the parameter run which is a public/global and special variable 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, at each iteration, each resistor will have a random value within the range set by their minimum possible and maximum possible values (i.e. in \$[R(1-tolb), R(1+tolb)]\$ range). And this function takes the parameter run (as stated before, a special variable for LTspice, just like time) as input, so you need to manually increase it at reach iteration so that the random generator doesn't generate the same value.