Those are your peroneal tendons. You have 2 of them. One runs along underneath the ankle bone and attaches at the 5th metatarsal. The other one runs under the arch and attaches on the medial side. A burning sensation or when you're riding or an ache afterwards probably means you're pissing these tendons off and are starting to develop tendinitis there. If it gets bad, you may find that your 5th metatarsal hurts badly, like a foot stress fracture, or you may feel some pain localized around the fibular head. You may also notice that the tendons protrude more under the ankle bone on the affected side than on the other, because they are tight.

I find that massage works the best, followed by ice or a cooling rub. I developed it on one side, and it turned out to be a secondary problem. The primary issue was my pelvis alignment on that side and how that was affecting my ITB. No wonder the lateral pain and stiffness eventually moved all the way down to the calf. Stretch your hips, glutes, and ITB regularly. Then treat the calf with self-massage. Look at your position on the bike too. Are you sitting crooked at all? Does the knee on that side track more inward? Do you point the foot more outward? Is there any reason you'd be shortening and flexing those tendons more on that side than the other? Occasionally, it can be aggravated by tightness in the tibialis anterior muscle (the one at the front of your calf, just to the outside of the shin).

If nothing helps, go see a physiotherapist. He/she will check your alignment, look for tightness higher up, check your gait, and will treat the tendinitis with massage, ultrasound, ice, etc. plus strengthening exercises if needed.