They had me do

In that context "had" functions as a causative verb. As a causative verb, "had" expresses:

a) I used a person to do something for me.
EX: I had Max WASH the car. (I had someone DO something)

b) I had something done by someone else.
EX: I had the car WASHED. (I had something DONE)

Some causative verbs take DO, whereas others take the infinitive TO DO. The verb "help" takes both. You have to memorize them, sorry.

let x DO y
make x DO y

allow x TO DO y
permit x TO DO y
force x TO DO y

help x DO y
help x TO DO y