One of the benefits of a whole life insurance policy is that is has an accrued cash value. In addition to the face value, which is the amount the policy guarantees to pay out if you die, cash value is the accrued value of the policy above the face value. Cash value accrues as you make premiums payments, and then earns value over that amount. It is this accrued cash value that can result in receiving a notice that your policy is "paid up."
When the policy is paid up, it means that you are not required to make premium payments for a period of time. Instead, the insurance company will deduct the amount of your premiums from the accrued value. The policy is not really paid up in the strict definition of the term, but it is capable of making its own premium payments. Depending on the type of policy and how well it has performed, you may have to make premium payments again in the future, or it may reach a point where the premiums are covered for the rest of the life of the policy.
You do not have to stop making premium payments just because the policy is paid up. If you do make payments, then the cash value will continue to increase, sometimes at an even faster pace, but usually only by the amount of "extra" money being paid in. For some people, allowing the cash value to build is a good idea because it will take the strain of paying life insurance off the policyholder in later years when they may not have as much money to pay out.
Another reason to keep the accrued cash value in your account is that you can borrow against that value. For instance, if you have accrued a few thousand dollars in cash value, you can make a loan to yourself out of the accrued value to make home repairs or perform other functions. If you do not pay back the loan, then the value of the policy will be reduced by that amount plus interest before the policy is settled.