The plastic or metal aglet at each end of a shoelace stops the shoelaces from unraveling. It also makes lacing the shoe far easier. If you have ever laced a shoe without an aglet, you will know exactly how difficult it can be. As soon as the frayed end gets caught on the edge of one of the holes you find you are pulling only a part of the lace through and the rest is unraveling. The trick is to moisten down the frayed end. Then can you lace a shoe with a broken aglet.
Wet the area where the aglet used to be.
Twist the end of the lace where the aglet used to be until all the frayed parts have stuck together.
Thread the lace carefully through the shoe holes. It doesn't matter which end of the lace you start with as long as you remember to thread the end without the aglet through the holes carefully.