What does +1 Proficiency mean?

Though it displays as a whole number, under the covers your character's stamina can have decimal values. Levels in applicable skills reduce, by a fractional amount, how much stamina it takes to use the tool. Here are the ones I was able to figure out.

These skills cause you to use 0.1 less stamina per skill level with these tools:

Foraging

• Axe

Fishing

• Fishing Rod

Farming

• Hoe
• Sowing a seed
• Watering Can

Mining

• Pickaxe

These always use 4 stamina:

• Milk Pail
• Shears

(Source: I used a decompilation tool to peek at the source code.)

I'm not sure about anything else, but higher Proficiency decreases the amount of energy usage for tools. I haven't upgrade my base Hoe at all but, without any hard numbers, I can say that I can till more spaces for the same amount of energy than when I started out.