Nobody pays tax on the first £10,000 of their annual income.
However it would clearly be daft to deduct no tax for month after month (until £10,000 had been earned in that tax year) and then to knock 20% off everything earned for the rest of tax year.
So the system allocates an allowance of £833.33 per month (or £192.31 per week, for weekly paid employees) as 'tax free' by dividing £10,000 by 12 (or by 52, as appropriate).
If someone doesn't earn more than that throughout the whole tax year then they'll pay no tax.
If they initially earn more than that they'll pay tax on the excess but they'll get it back later if their income then drops so that their total pay for the year is less than £10,000.
If they earn very little during the first part of the tax year they'll effectively be 'in credit' for tax purposes if they start to earn more later on, so they'll still not have to pay any tax if their total pay for the year is less than £10,000.