You will have to make sure that the folder structure of the RAW files is the same as on the old system. Then you will need to find the copy of your catalogue file. It will have a name in the format My_catalogue.lrcat. The easiest way to find it is to go to the Edit tab on the toolbar in Lr and select Catalogue settings. In the General tab it tells you the location, and file name of the catalogue. If you click the show button it will take you to the folder in Windows Explorer. Or you could open Windows explorer and search for *.lrcat.
You will need to transfer that file to your new computer and start using it as your catalogue. If you start with Lr closed and double click the transferred .lrcat file you will start using the new catalogue. By default Lr is set to use the last catalogue, so after that you can start Lr in the normal way. If the folder structure for the RAW files is identical to the old system Lr should now just show you all of the images with the preexisting edits as well as any metadata you added in Lr.
If you get image missing notices, then your folder structure is different, or they are starting in a top level folder which is in a slightly different place. If it is just the top level placement that is different you can easily fix this, click on the option to find image files and navigate to the top level folder for your images. Find the image file that matches the file name for your current image exactly and select it. There should also be an option to find nearby files. Ensure that this is also selected. If the rest of the folder structure is the same as the old computer it should find all of the images. If not you will have to do this for each folder that is in a different relative place to where it was on the old computer.
When you install Lr on a new system, or even if you have to reinstall it, part of the process is to create a new catalogue. So you will have to ensure that you start it with the old catalogue to see your existing work.
I'm pretty sure I warned you about needing to do this as part of the transfer process. Or at least you should keep the RAW files in the same folder structure, and to move your existing .lrcat file to the new machine as part of the transfer process in your other thread.