After installing and running it, with the game solitaire running, click on the
"Select a process" button on Cheat Engine.

A process list will be opened, and you`ll have to select the process of the game you want to cheat (In this case solitaire.exe is the process you are looking for).

If needed you can find out the name of the process by pressing "Ctrl+Alt+Del" and selecting Task Manager, and then on the applications tab, right-click the name of the game and then select "Go to process".
With the right process selected on the process list on cheat engine, play some solitaire until you score get a score different to 0. As soon as your score change on the game, go to cheat engine and type the new score on the value box and click "First Scan" (You might need to do this fast, because on solitaire you lose score as the time passes).

After scanned you`ll notice alot of addresses and values on the left box, one of which is the value you want to change. Just wait a few seconds or play a little for the score on the game to change, and then type the new value on Cheat Engine and then click "Next Scan". If you did everything correct you`ll see only a few addresses on the right box (possibly only one), so keep inputting the new value on cheat engine and clicking Next Scan until you end up with only one address.

Then double click that value and it will be sent to the box on the bottom. All you have to do now is to double click the value on the box below and just change it to wathever number you like. Your score in solitaire will also be changed to that value, simple as that.

Note that you can use only cheat engine and similar softwares to access data being processes in your computer, so accessing data of online games isn`t possible. For the record, it is still possible to cheat flash games, as most of them runs in your computer, so you can change the score and then submit it online.
If you want to learn more about Cheat Engine you can always try the tutorial that comes with it, which is great.