Why does it take longer to cook food at high altitude?

Because water boils at a lower temperature when air pressure is lower, and boiling water is the ceiling on how hot anything cooked in it can get.

A liquid boils when its vapour pressure matches the pressure of the atmosphere pressing down on it. At sea level that happens at 100°C. Air pressure falls with altitude, so the balance is reached sooner — roughly 1°C lower for every 300 metres of elevation. At 1,500 metres water boils at about 95°C; in a high mountain town at 3,000 metres, around 90°C.

Once water is boiling, adding more heat produces more steam, not a higher temperature. So food simmering in water at altitude is being held at 90°C rather than 100°C, and chemical reactions, including the ones that soften starches and break down connective tissue, run substantially slower at lower temperatures. Pasta, rice, pulses and boiled eggs all take longer, and dried beans can take dramatically longer.

A pressure cooker solves this directly. By sealing the pot it raises the internal pressure above the local atmosphere, pushing the boiling point back up. It is genuinely more useful at altitude than at sea level.

Baking is affected differently and more awkwardly. Lower pressure lets leavening gases expand more easily, so cakes rise faster and can collapse before the structure sets, while liquids evaporate more quickly and batters dry out. High-altitude baking adjustments typically involve slightly less leavening, a little more liquid, and a modestly higher oven temperature.

Frying and roasting are barely affected, because oil and hot air are not limited by water's boiling point in the same way.