There are not many countries in the world that have clearly modeled their entire constitutions on the Constitution of the United States. The article in the link below says that constitutions used to be more similar to the US Constitution, but that the similarities have declined over time. Fewer countries now base their constitutions on the US Constitution in any way.
There are certainly some countries whose constitutions are similar to ours in some ways. For example, the Philippines has a presidential system like we do, with a bicameral legislature and a national supreme court. There are differences between their constitution and ours, such as the fact that their president is elected for one 6-year term and the fact that their constitution guarantees the right to travel and live anywhere in their country. One of the smallest countries in the world, the Federated States of Micronesia, has a constitution that is similar to ours in that much of its “Declaration of Rights” is taken directly from our constitution. It also has a presidential system with a legislature and a supreme court, but its legislature is unicameral and the president is elected by the legislature. The constitution of Liberia is also heavily based on the US Constitution.
The one thing that all of these countries have in common is that they were in some way under US tutelage at important points in their history. Liberia was founded by freed slaves from the US. The Philippines was a US colony until it was granted independence after WWII. The FSM was a US Trust Territory for about 40 years after WWII. These countries were strongly influenced by the US and their constitutions show that fact.
To the extent that other countries’ constitutions have similarities to ours, the most likely reason is that the US has been the world’s most successful and enduring democracy. It would make sense that countries would see that our constitution has worked for us and would want to copy at least some parts of it. However, they have not wanted to copy all parts of it since they want to tailor their constitutions to their own specific national circumstances.