Seed dispersal's importance becomes more clear when you think about it in relation to competition for resources. If a plant were to drop all of its seeds right below itself, the seeds would be in competition for the very resources that the parent plant has already "won." The parent plant would deny the seeds access to sunlight because of its larger, more well established leaf and branch structuring. It's root system is also more robust, so it would deny any seeds access to water and soil nutrients. Even if the parent plant were not a factor, if all of the seeds were dropped in a single location, the same kind of competition would follow. Instead of competing with the parent plant, the seeds would be competing against each other for the same set of limited resources.
By dispersing the seeds over a large area, a parent plant gives the seeds a greater chance at finding space and untapped resources. With those advantages, the seeds stand a much better chance at survival. Plus it allows a species to begin "colonizing" a new area. Instead of simply growing in a very limited area, a species can extend its habitat and niche.
Additionally, organisms that feed on seeds (birds) tend to search for areas where there are a lot of seeds in a small space. If seeds are scattered further apart from one another, they stand a greater chance at not being consumed by birds.
Seed dispersal is the moving of seeds to a new location. This helps move genetic information away from the parent plant into new and uncharted territory. As mentioned in the previous responses, this prevents the plant from dying due to a lack of resources.
We can also look at seed dispersal as being evolutionary advantageous to the plant. Seed dispersal ensures survival of the plant species should an environment become uninhabitable due to construction, pollution, an alien species, etc. - At least there will be some survivals in a new location. This is called the Founder Effect in Biology.
Furthermore, when dispersed to new environments, plants have the opportunity to adapt to the location's conditions. These adaptations will also ensure the prolonged survival of the plant on Earth.
Seed dispersal is important so that the new plants have enough resources and do not compete with the parent plant. Fruits are a great way to disperse seeds. Animals eat the fruit and seeds and then release the seeds some distance away from the parent plant.
If all the seeds were deposited close to the parent plant not as many of them would survive to adult plants. Dispersing seeds increases the chance of having more offspring.