Some people can't understand things without a visual element. When you use graphs to solve a system of equations, you're showing the graph of the equation, which is a proportional visual representation of the figure that equation represents. This will give visual learners a better understanding than just seeing a bunch of numbers on paper would.
The graphing method for solving systems of equations is easily done using the "Graph" function on graphing calculators. Graphing each equation with a graphing calculator to find the solutions is much faster than using other methods. You might still be able to solve simple equations faster algebraically than graphing would, but graphing saves time with trigonometric and other nonlinear equations that would take a considerable amount of time to solve by hand.
Linear systems of equations can have one solution, no solutions or infinite solutions; however, more complex systems can have many solutions, infinite solutions, no solutions or one solution. When you're solving a complex system, graphing the system shows you the way the graphs interact. You're less likely to miss multiple solutions when you're able to see where the graphs intersect one another.
Identifying the type of equation you're working with helps you predict what the graph should look like when you graph it. Linear equations, for example, have no powers, no trigonometric and no logarithmic functions. Linear equations will form a line when graphed. Exponential equations have terms that are raised to a power. The type of graph it forms depends on the exponent; for example, when the highest exponent is 2, the equation forms a parabola. If you graph an equation and the graph doesn't match the type of equation you're using, you know you made a mistake.