The size of your arena is a measure of how important your team is. The bigger your arena, the larger your staff budget will be, meaning you can hire more staff members and better quality coaches (staff budget = Arena Level X $1M). Better arenas also come with extra drills. For example, a level 2 arena will give you one more drill than a level one arena would give.
How They Work
Each arena level has its own seating and establishment capacity. So bigger arenas allow for a higher fan attendance, which leads to increased credit earnings. Each arena also has its own establishment limitations (the number of beer stands it can hold, for example). The more establishments you have in your arena, the more fans will spend at your games, which leads to increased credit earnings. You will need to max out on your current arena's establishments before you can upgrade to the next arena level.
With a bigger arena comes bigger earnings, but also bigger costs. The price (in credits) of buying out players, hiring staff members, and using staff services is proportionate to your credit earnings. Although higher credit earnings are somewhat canceled out by higher costs, upgrading your arena is still well worth it to get the increased staff budget and extra drills.
The Arena Level 5 Rule:
If you have a level 5 arena and you made less than 800,000 credits in your last home game, for cost purposes, your last home game's income will be treated as the greater of 1.5% of your team's total credits and 15,000 credits.
If you have a level 5 arena and you made at least 800,000 credits in your last home game, your last home game's income will be used for cost purposes. Cost purposes just refers to the basis for determining how much you have to pay (in credits) for things like staff, coaches, scouting, and buyouts.