When you develop your film, the thinnest part of the negative (corresponding to the shadows) develops first, and then as time goes on, doesn't change very much. The parts of the negative which had the greatest exposure - your highlights, will get denser and denser as your development time increases.
"Exposing for the shadows" means that you determine your exposure by metering in the shadow area - and this part of the negative will come out right pretty much regardless of how long you develop the film.
In terms of figuring out what the exposure should be - you normally want your shadows to be about 3 stops down from the mid tones in the image - your camera meter is set to make the mid-tones come out in the middle, so if you get close to the shadows with your camera, and take a meter reading, the exposure that you will get will be what it takes to make that part of the image a mid-tone, but that's not what you want - you want the shadows to be dark, so you want to expose less by 3 stops. One way of doing this is to set the ASA on your camera 3 stops faster than the film - so for TRI-X, set your camera ASA dial to 3200 - then either use a long telephoto lens or get up close to the subject so that the shadow area fills the view finder and take your meter reading. This will give you an exposure where you should have sufficient density in the shadows to keep some detail there.

The second part of the adage is "Develop for the highlights" The longer you develop the film for, the more density you will build in the highlights, and the greater the contrast of your final image. Since you are shooting roll film, you will have to treat the whole roll of film the same. If you are taking pictures in a bright sunny area, where there are deep shadows and very bright highlights, you are going to have more contrast than you want, so you will want to develop for a shorter amount of time than usual. If you are taking pictures of sand, under overcast skys so there is very little contrast in your image, you will want to develop for a long time to increase the contrast.

TRI-X is pretty forgiving, so you should be able to get pretty close with just metering the whole scene. What you may want to do is to set the ASA to 3200, meter on the shadow area, and let's say that it says to expose at 1/60 and f11. Then set the ASA to 400, step back and meter the whole scene - you should get something close to 1/60 at f11 unless there is something really strange about what you are metering.