Some commentators say that it is because of the demand to "turn away your foot", mentioned in the same verse.) This prohibition is suspended in the face of a mitzva. "Rebbe Zeira said, When I would see the scholars running to shiur on Shabbat I used to say, These scholars are desecrating the Shabbat! But since I heard Rav Tanchum say in the name of Rebbe Yehoshua ben Levi, A person should always run to hear the law, even on Shabbat, as it is written 'They will go after Hashem like a roaring lion' (Hoshea 11:10), I run too." (Berakhot 6b.)
What kind of mitzva is important enough? The Semak writes that it is permissible to run even for Shabbat delight, oneg Shabbat. This includes games that involve running, or running to see something interesting. And so rules the Shulchan Arukh: "Youngsters who take pleasure in jumping and running, it is permissible. And also to see something they enjoy". (SA OC 301:2. This definitely does not include jumping and running for training, as stated explicitly in Tosefta Shabbat 16:22.) This includes also playing ball.
While the Shulchan Arukh rules that playing ball on Shabbat is forbidden (SA OC 308:45), the reason is not the game per sebut rather the concern of forbidden melakha, and in any case the Rema writes that the custom is to be lenient.
So far it seems that playing games on Shabbat is not only permissible in itself, but is actually a mitzva, one that can permit running, which is otherwise forbidden. However, as the Beit Yosef points out (OC 308), there is another prominent source bearing the opposite message. The Yerushalmi on Taanit describes in great detail the destruction of Beitar in the time of Bar Koziva. (Today he is often called Bar Kokhva, but to the best of my knowledge the Talmud never uses this name.) Why was it destroyed? One of the explanations is that they used to play ball. Most commentators explain that they played on Shabbat, as mentioned explicitly in Midrash Eikha Rabba (on Eikha 2:2).
One explanation is that they used to desecrate Shabbat in their games, but even this explanation seems to single out game playing as particularly blame- worthy, since presumably there were other kinds of Shabbat desecration in the city as well.
I haven't found any source that explicitly deals with this paradox, but from a few commentators it seems that an important distinction is the element of competition. A number of Rishonim explicitly mention that running and jumping are permitted "because they don't do so in order to earn". Apparently professional sports are nothing new and even in the time of the Rishonim it was common to put money on sports competitions. The Bach on OC 301 rules that the permission to run and jump for fun is only bediavad for this reason.
This also seems to be implicit in the words of the Shulchan Arukh, which permits youngsters who enjoy jumping and running, not youngsters who enjoy winning. Competition for its own sake contradicts the spirit of Shabbat, which is entirely a spirit of harmony and cooperation.
Playing sports is a positive occasional pastime for active youngsters, as long as the enjoyment is in the activity itself. But when the competitive urge becomes dominant, and the physical activity secondary, this is an inappropriate delight, one that is not considered an oneg Shabbat and doesn't justify haste on Shabbat.
Rabbi Meir's book Meaning in Mitzvot is now available as an e-book through Amazon, ibooks, Google Play and B&N.