Why does lowering interest rates boost aggregate demand?

Lowering interest rates boosts aggregate demand through a process known as the transmission mechanism. We can make this easier to understand by splitting AD into its component parts and examining each one.
AD = C + I + G + (X-M)
When interest rates are lowered, this effectively reduces the cost of borrowing money. You have to pay less on top of the amount you're borrowing. For consumption (C) this means people can afford to buy expensive new durable goods. For example it's cheaper to get a loan to pay for a new car or a new fridge, both of which are consumer goods, and so lower interest rates increase consumption. Cheaper borrowing also applies to investment (I). Firms tend to borrow vast amounts to purchase new equipment, say to open a new factory. With interest rates lower, the cost of opening this plant is reduced which makes it more profitable and so more firms invest. A similar approach applies to government spending (G) governments can borrow money to spend (rather than tax which takes money out of the economy) at a cheaper rate so are incentivised to do so.
Now the case for exports minus imports (X-M) is a little more complicated as it requires an understanding of how exchange rates work. However we can use a simple chain of reasoning. The lower interest rates are, the less money savers get for saving money in UK banks. Now this is significant when considering "hot money flows", extremely large pools of savings from international institutions and individuals that look for the best return on saving their money. If interest rates were reduced in the UK, they would get a lower return for saving there and would move their money elsewhere. This reduces demand (diagram helpful here) for converting money to the British pound, lowering its nominal value and thus depreciating the exchange rate. A lower value for the pound makes it a less expensive currency (in terms of other currencies) and as such UK prices are cheaper to foreigners making UK goods more attractive as exports. The reverse happens with foreign goods from a UK perspective, therefore imports become less attractive. As result exports increase (X) while imports decrease (M) so (X - M) gets larger in value.
As a result of all these comments increasing, the value of AD is increased.