In an "if-then" statement in math, the "then" part of the statement is the conclusion. It is the part of the statement that is the end result.
In geometry, a proof is written in an if-then format. The "if" is a condition, and if that condition is true, then the second part of the statement is also true, which is the conclusion. It works like any if-then statement. For instance, the statement "If an animal shelter has only cats and dogs and 35 percent of animals in the shelter are dogs, then 65 percent of the animals in the shelter are cats." The conclusion of this statement is that "65 percent of the animals in the shelter are cats."