Some good explanations from reddit (lightly edited):
Why SSRIs work for anxiety
When you are confronted to a stressful event, your brain anticipate the energy and focus for the event and gonna stimulate (or slow) different systems in your body. In stress, you are gonna use your accelerator (sympathetic system) and pump more blood, breathe more oxygen, get more focus, and prime your muscles. This is you getting ready or reacting to an event.
In an anxiety disorder, you have your foot constantly on the accelerator — more situations are perceived by your brain as stressful. This includes non-threatening situations, as well as overreacting to smaller stressors.
Antidepressants regulate some chemical pathways of the nervous system. It forces the accumulation of certain chemicals in your brain so these pathways are not constantly stimulated. It may helps to manage the disorder better.
Why people are anxious
Imagine everybody’s anxiety levels on a scale of 1-10. Imagine that anything in a 3-5 is a fairly manageable range, 6-8 is where you are freaking out, and 9-10 is catastrophic.
In general, those without anxiety live in the 0-2 range of anxiety on a daily level. When an event occurs, their anxiety levels jump 1-3 points, so it is manageable. If the event is a major event, their anxiety levels can jump to maybe a 5, possibly a 6 or 7 on the scale.
Someone with anxiety, on the other hand, lives at a 3+ range daily. If an event occurs, their anxiety levels can jump 1-3 points which can put them anywhere between manageable into the freakout range. If the event is major, their anxiety jumps more points, which can put them in a generally higher range on this scale.
Anxiety medication can help people who live in the 3-4 daily be instead, in a 1-3 range on average. It doesn’t sound too major, but can make all the world of difference to someone who is living in a constant freak-out range every day of their life.