Why go SURFACED?

Those U-Boats could only sail submerged for a few hours, and they were not exactly quiet so Der Alte's plan was to submerge just before the patrol line and then drift through the strait not using the engines - the Strait og Gibraltar has a rather strong eastward surface current.
It was a good idea. But, as the LI says when they sit trapped on the bottom of the Strait, 'it also has to come off'. For instance he could not be sure exactly where the Royal Navy's patrol lines would be.

The surface speed of a german VIIC U-Boat was around 18 knots the submerged speed was only about 8 knots. So, the first reason was the speed.
The british sonar system would most likely locate the Boat either submerged or emerged. And also it was likely, that there would be mine fields specialy designed against submerged U-Boats, wich were at least as dangerous as the british canons at the surface.

Emerged is the wrong word. Subs are submerged, or SURFACED.
Surfaced is the word you're looking for in your post and title, not emerged.

There are many reasons they ran the strait SURFACED rather than submerged.

1) ENDURANCE:
A Submarines endurance was much more limited when submerged. At full speed they were noisy and easily detectable on British hydrophones, and were still much slower (about 8 knots max) and their batteries would be drained flat within about an hour, forcing them to surface no matter what. Even if they ran it a a very slow speed, they still ran the risk of being detected by hydrophones, it would take them far far longer to complete the transit increasing the odds of detection, and another factor which makes it an impossibility which I will address later.

2) TACTICAL AWARENESS:
A sub on the surface at night is very very difficult to spot even by an enemy aware of and looking for the sub. If they submerges they ran the risk of being detected by Hydrophones and pinpointed. On the surface, British patrols had to visually spot them in the dark and foggy low visibility. A Sub on the surface however has a better chance of seeing and avoiding surface patrols by the enemy by spotting and altering course to avoid long before patrols had a chance to spot them in return. Submerged,a submarine is virtually blind but for their own sonar which would not give them a very accurate picture of the surface situation. this increased the odds they would blunder into a patrol and be detected.

3) PHYSICS:
There is a permanent current near the surface that runs Eastward into the Mediterranean Sea. This is coupled with a Deep water current that runs Westward out of the Med.
Had they gone deep and tried to get through the strait, they would have been fighting against this deep and strong westward current as they tried to move east through the strait. At a slow pace and low battery usage it is impossible as the current would be stronger than the sub's motion and they would have been carried West as well. Had they gone to full power to make headway against the current, they would be noisy as I pointed out above, and their battery would have been drained flat in no time, long before they could have made it through the strait.


I joined the Navy to see the world, only to discover the world is 2/3 water!

You should read the book. It's all explained there in details. As others have already answered, they couldn't stay too long submerged and traveling under water was much slower and consumed more fuel.