How does water evaporate if it doesn't boil?

Evaporation is a different process to boiling. The first is a surface effect that can happen at any temperature, while the latter is a bulk transformation that only happens when the conditions are correct.

Technically the water is not turning into a gas, but random movement of the surface molecules allows some of them enough energy to escape from the surface into the air. The rate at which they leave the surface depends on a number of factors - for instance the temperature of both air and water, the humidity of the air, and the size of the surface exposed. When the bridge is 'steaming': the wood is marginally warmer than the air (due to the sun shine), the air is very humid (it has just been raining) and the water is spread out to expose a very large surface area. In fact, since the air is cooler and almost saturated with water, the molecules of water are almost immediately condensing into micro-droplets in the air - which is why you can see them.

BTW - As water vapour is a gas, it is completely transparent. If you can see it then it is steam, which consists of tiny water droplets (basically water vapour that has condensed). Consider a kettle boiling - the white plume only occurs a short distance above the spout. Below that it is water vapour, above it has cooled into steam. Steam disappears after a while, as it has evaporated once again.

For every temperature, there is some amount of water vapor that can exist as gas mixed in with the air. This is called the saturation pressure of water at that temperature. The relative humidity is the amount of water vapor pressure, expressed as a percentage of the saturation pressure. As you increase the temperature, the saturation pressure increases.

Steam is water in its gaseous phase.

You can't see water vapor, you can't see steam, but you can see mist, which is liquid water droplets suspended in the air.

When you boil water on the stove, you get steam. This then cools when it comes into contact with the air, increasing the relative humidity above 100%, so the water vapor condenses into mist.

If the relative humidity is bigger than 100%, water vapor will condense from the air, becoming dew and/or mist. If the relative humidity is less than 100%, water will evaporate into the air, becoming water vapor.

If the wooden bridge is warmer than the surrounding air, and the relative humidity is around 100%, then water will evaporate off of the wooden bridge, turning into water vapor (the relative humidity is lower right next to the bridge, because the bridge is warmer). When the air containing this water vapor rises and cools, water condenses out of it, turning into the mist that you see.

Here is a graph of the saturation pressure (from this website). Note that at 100°C, the pressure is $\approx10^5$ Pa $=1000\,$hPa, which is roughly atmospheric pressure. This means that at 100°C, you can have pure water vapor at atmospheric pressure. This is why water boils at 100°C at sea level—a bubble of steam can form below the surface of the water. At higher altitudes, the boiling point can be substantially lower.

Below "boiling point" (not always 100C), water can exist in both gas and liquid phase, and has a temperature-dependent vapour pressure, which represents a point of equilibrium between liquid water wanting to evaporate and water vapour wanting to condense. When liquid water meets dry air, it is not in equilibrium; water molecules evaporate off the surface until the amount of water in the air creates enough vapour pressure to achieve equilibrium.

When water is heated to a temperature of 100C, the vapour pressure equals that of sea-level air pressure. Since the air pressure can no longer overcome the vapour pressure of the water, the water boils.

At higher elevations, air pressure is lower; as water is heated, its vapour pressure overcomes ambient air pressure at a lower temperature i.e. the boiling point is lower.

Vice-versa for higher pressures.

As for the steam rising off the bridge, that is actually water vapor condensing. Very close to the wet surfaces, the air is saturated with water vapor, which is transparent. It is also less dense than dry air, so it rises. As it rises away from what is likely a warm surface, it cools, As it cools, it condenses, but it is also mixing with more drier air, so it evaporates again and disappears.