When a team, let's take Toronto Maple Leafs as an example, play in a home field with all their fans rooting or cheering them on, the team gets very pumped and strive to do their best because they have supporters with them. Also, the team would be familiar with the stadium, and the home field wouldn't boo the home team because they are the ones that the home field are voting for. On the contrary, the away team would get booed because they are not the ones the fans want to win. If Toronto Maple Leafs go against Boston and they are playing in Toronto, Boston would get booed and Toronto would be cheered on.
The home team will always have an advantage on the home field because they are more familiar with the area and know its idiosyncrasies. In addition, the home team has the advantage of the crowd rooting for their success, and the psychological factors involved with that cannot ever be discounted.
Home field advantage is a real thing and exists in many sports. If a home team's crowd is really excited, they can get into the opposing team's head and distract them. There's a lot of teams that have great home records because of home field advantage. Referees tend to give a few calls to home teams as well.
Yes, the cheering from your team might be encouraging. However, the booing will make the team play better as they will want to beat the home team which is the equivalent of booing back at the crowd. The pitches are almost always flat and 'learning the pitch' is a thing of the past.
There are some teams where it is a very big deal, they are able to settle into a routine they are able to handle but for some teams they are strong enough that they are able to win no matter what field they play on. Its just a matter of the team and the effort put out that day.
With the exception of a few select stadiums, home field advantage is no longer a factor. This changed with the advancement of radio technology in sports, particularly between the sidelines and the players. This, along with the revolution of travel has allowed stadiums to be filled with equal fans alike. Weather no longer plays an issue in the NFL either, as it is now a pass first league. The frozen tundra doesn't benefit Aaron Rodger's passing game, as a more recent example.