You need to compile your map. Here are the steps:

Go to File > Run Map...

This dialog box will appear. This gives you options on how to compile your map. If you're going to publish your map, I recommend leaving everything on "Normal". Here's how everything works.
Compiling has 3 main steps: VBSP, VVIS, and VRAD, represented in this dialog box as "Run BSP", "Run VIS", and "Run RAD". VBSP turns all of your level geometry into a base .BSP file that can be run by a game. It is the only part of the compile process that is mandatory to produce a working map. VVIS performs optimization tests. It checks what parts of the map can be seen from other parts of the map. It also does a few other things, such as making water appear (if your map has any). VVIS will not run if you have a
leak in your map. VRAD adds light to your map by casting light from any placed light entities. Not running VRAD causes everything to appear super bright, which looks ugly. It's fine for private test compiles, but if you're getting ready to publish your map, definitely include lighting. VRAD will produce weird lighting if VVIS doesn't run, but you can trick VRAD into thinking VVIS has run by setting VVIS to "Fast". After everything is run, the compiler will launch TF2 and immediately put you into your map. (Note: TF2 must be closed for this, otherwise an error message will appear.)
You should basically always run VBSP in normal mode- don't change that. VVIS and VRAD are optional. If you're just looking to do a quick compile to test a part of your map or see it in-game, you can turn VVIS and VRAD off, which will make your compile faster, or you can run them in "Fast" mode. "Fast" VVIS doesn't actually perform optimization, but it still does some other stuff, such as making water appear and allowing VRAD to do proper lighting. "Fast" VRAD produces sloppy lighting, but it's useful for quickly testing your lighting. Setting everything to "Normal" will take the longest, but it is highly recommended for when you publish your map.
The "Don't run the game after compiling" checkbox will do exactly as it says- it won't run your map in TF2 after compiling is completed. If you want to see your map in-game after compiling, leave this unchecked. The "Additional game parameters" is a set of parameters that will be run when your map runs in TF2. They are not required.
To compile your map after selecting your settings, click "OK".
EDIT: Looks like you already did that.