Keep in mind you have (2) problems when you disconnect a drive and try to attach it to another system:
1) getting the data bus wired correctly
2) powering up the drive so the platters spin and the heads move.
You must understand whether your adapter covers only number 1) above.
But, if you already have a complete enclosure, it should provide both a data path and power to the drive you insert.
Does your enclosure have only a USB cable coming from it? I would guess that
it also has some power going to it as well. Otherwise you will only be able to operate on rather small amount of power provided by your computer's USB port, insufficient to spin up a drive.
The thing to understand here is that you should be making (2) connections
to the drive itself, 1 for data, 1 for power. This may not necessarily mean that there are 2 plugs - there are connectors/drives where data and power are combined into a single strip of pins.
Then it is up to whatever OS is on the computer you are using as the repository for your recovered data to recognize the USB <-> ATA/IDE/SATA device and the HFS format on the drive.
I have done this before, but its been a while. The USB <-> ATA/IDE/SATA adapter is confusing things and adding an extra layer of complexity, but sometimes thats the way it goes.
hope this helps to some degree