тархи - brain
толгой - head
тархи толгой - collocation. Depending on the context may mean either, brain or head.
тархи толгой гашилгах - idiom. Meaning: to think a lot (in order to find our some solution).
толгой эргэх - dizzy, feel dizzy
Ганц энэ асуудлыг ойлгох гэж би тархи толгойгоо эргэтэл бодов.
Literally: In order to understand this one problem, I thought so much about it that my head got dizzy.