When I look at a distribution with higher excess kurtosis, it shows it as more peaked. I am assuming that this means the distribition has more values close to the mean. However, they also say that this means more values are located on the edges (extreme ends) of the distribution as well. How come a distribution with high kurtosis does not show this in visualization. It just shows a high peak with smaller outsides.

When a distribution has positive excess kurtosis, it has a higher peak than a normal distribution (with the same variance), and thicker tails than the normal distribution; it has lower probability in between the peak and the tails, say, between about 1σ and 3σ, and between −1σ and −3σ.

But how come when I could at a picture of a frequency distribution that has positive excess kurtosis, these largers tails are not shown?

Because the tails are incredibly thin.