I don't understand why it is necessary to use a trigger on an oscilloscope for data acquisition
You can demonstrate the problem very easily on most scopes by setting the scope trigger to &quot;free run&quot; and setting the trigger level above the peak of the test waveform. (You can usually use the scope's calibration point squarewave for this test.) The scope will run a trace, wait a while for a trigger signal, give up and run another trace starting at a random point. The result will be a blurred mess with the waveform barely visible.

Figure 1. Trace of a sinewave without triggering. Image source: Test and Measurement Tips.

Since it repeats you'll get the same thing, right?

Yes. You get the same thing over and over at different starting points. It becomes very difficult to make any measurements.

Figure 2. The same trace with correct trigger setting.

From the comments:

Yes I've certainly seen the first screenshot as a result of incorrectly triggering, but I don't see why that happens. It's showing different voltages at different times, when in reality there is only a single voltage at a given time.

There's only a single voltage at a given time but the scope re-uses each x-axis point over and over again because the display isn't infinitely wide.
On a digital 'scope the blurring happens due to persistence of vision. Your eyes and brain can't extract just one scan or screen update. On the other hand, if you take a photo of the screen you will get just one trace if the exposure time is short enough.
On an analog cathode ray tube 'scope you have the additional factor of the tube's phosphor coating decay time. This had the effect of making the most recent traces appear brighter as the older ones faded away.