Why do ISPs change your IP address?
When ISPs were first starting, everyone connected to the Internet over a modem. And most people used the Internet for a few minutes to a few hours per week. Assigning a static IP to every subscriber would have been very expensive, for something that most people used just a few minutes a week.
As broadband connections have become more common, the practical reasons for not assigning a static IP have become much less noticeable, as now the majority of connections are "always-on"--even when nobody is (actively) using the Internet.
So there's a bit of a historical reason not to use static IPs--customers are already accustomed to using dynamic IPs.
When modern ISPs enforce dynamic IPs these days, it may be in part to distinguish between "consumer" and "professional" services--by reserving static IPs for customers who pay more, it gives customers who need that feature an incentive to upgrade their service level.
It can also serve as a deterrent for people abusing their consumer-grade service. Many ISPs, for instance, explicitly prohibit running "servers" on a home Internet connection. If every home user had a static IP, they'd be more inclined to abuse such terms of service.
It's also less of a management problem to assign customers dynamic IPs. If you move across town (but within the same ISP's service area), there's no need to re-assign how your static IP is routed; you'll just get a dynamic IP that exists in the new neighborhood.
I can only speak from the point of a system administrator with a small to medium sized network. But this is a complicated topic so I am going to try to answer in the simplest fashion.
The goal of using dynamically assigned IPs is that most clients do not need the benefits of having a static IP. In your question, connections that would need a static IP would be customers that are hosting services like web servers, email, remote desktop, VPN, etc. The reason for this is that in order for machines to find them on the Internet, they would need to be able to track back to that IP by looking up the IP via DNS. For most home customers, there is no need for this so there is no reason to assign a home user a static IP.
The IP lease time is set by the ISP and typically if your device is still online and requests a new lease, the ISP will give you back the same IP. As nothing is gained by giving you a different IP. This also lets the ISP move blocks of IPs around and do other maintenance without interupting services. Most routers/modems have a display for the WAN IP and typically will display the lease time. It is usually a few hours to a few days. Odds are, your modem asks for an IP regularly, but the DHCP server simply hands it back the same address. When it has a reason NOT to, your IP will be updated with whatever new IP is being handed out.
Static IP addresses are much more difficult to manage. Even most "static" IP addresses are assigned dynamically through DHCP using static reservations, but with either a statically configured IP address or a DHCP static reservation, if the client connection device changes (hardware replacement, upgrade, etc), then someone needs to make a change either on that device or in the DHCP configuration. Dynamic assignment avoids this problem.
Using DHCP also simplifies the network design and allows for easier changes later. If the ISP chooses to make changes to the way they are using their IP addresses, they can do so without making manual changes on client connection devices. Examples of such changes could include any of the following (or others):
- Increasing the size of the network (moving from a /24 network to a /23)
- Decreasing the size of the network (moving from a /24 to a /25)
- Changing the network in use entirely (moving from 10.0.0.0/24 to 192.168.0.0/24)
- Changing the gateway IP address
- Changing the DNS server addresses
DHCP also allows for the ISP to provide additional configuration to some devices. For instance, if they want to provide images/configuration on a TFTP server, this can be provided to the device through DHCP.
Ultimately, it is much easier to manage and provides less complications to administering the network.