The reason why it would appear complicated would have to be that simply you're not used to the system. It makes sense to me in that I know how it works and I see no issue with it. So the system itself has nothing wrong with it, the people understood it, not just in the UK but throughout Europe where other countries used the same/similar system, such as the Dutch Guilder (divided into 20 stuivers, 16 penningen/stuiver (320 penning for each guilder)). It wasn't decimalised until the Guilder was reintroduced after liberated from Napoleon.

So pound sterling was split into £.s.d (pound/shilling/pence) with 20 shillings to the pound and 12 pence per shilling as you've said, so 240 pence to the pound. Come 1971, it was finally decimalised like the other European currencies, with the shilling now representing the decimalised value of the modern-day five-pence coin (old coins were used after decimalising, with shilling coin(s) being valid until the 1990s).

The coins of the pound were based on silver, with one pound of silver being split into 240 subunits of pennies.

I just refreshed the page to see that naryn has also commented on this and added extra details and providing tradition as an explanation. Along with the usage of 12, which is present strongly in the pence system. 12 pence to the shilling, 20 shillings to the pound, 12 divides well into the pound through 240 pence.

For an extra bit of information, following the practices of banking in providing interest based on my dad's experiences when he worked at the bank through training, along with books about interest on sums of £.s.d and other training stuff from Lloyds, to work out the interest for bank accounts, the sum was done in its pence form. It is difficult in itself to work out what 5% interest on £1 13s 7d would be at face value, and so you would convert the amount into pence then work it out like that (this was also something taught in schooling, as they still do with percentages).

£1 = 240 pence
13s = 156 pence
240+156+7 = 403 pence.
5% of 403 = 20.15 pence.
20.15 pence is the interest you would gain at five percent.