Does higher dBi mean further range, for an antenna?
Yes, you will get more signal strength from a 7 dBI antenna than a 2.2dBI (specifically 4.8 dB). It solves that by radiating energy more directionally than an idea antenna that radiates evenly in all directions (0 dBI).

This increased signal strength of 4.8 dB is 10^(4.8/10) = 3 times more power. That will increase your range by about 70% in ideal conditions.

Since it is directional, you will need to point it more carefully. Specifically the linked antenna is pretty much a vertical wire. It radiates in a circle around the antenna; your receiver shouldn't be much above or below this plane.