Mainly by brute force. The grandmasters could use their knowledge and experience to eliminate weak lines but basically they evaluated every move and possible response for several turns. If there weren't too many candidate moves and responses they could end up analysing even further.
When I was first learning the game I used to look at every possible move and every possible response to help keep myself from dropping pieces. Of course, I was only doing that for one move. And I still dropped pieces.
Does anybody have a clue how games were analyzed when computers didnt exist. How would chess players like bobby fischer know if games he analyzed were any at a advantage or disadvantaged or straight equal?