Can someone explain to me what is happening here?

I want to calculate the total distance between an unknown number of cities that is within an array. For this I have to create a function that calculates this distance.

function [ route ] = pcv_stub( matDistance, source)
  %matDistance is the matrix with the distances between each city.
  rng(0)   
  n = length(matDistance);
  route = source;
  cities = 1:n;
  cities = setdiff(cities, source);
  while length(route) < n
    city = randi(n);
    if ~isempty(intersect(cities, city))
        route = [route, city];
        cities = setdiff(cities, city);
    end
  end
end

function total_distance = total_dist (route, matDistance)
   %matDistance is matrix distance
   vector_distance = [ ];
   for i = 1:length (route) - 1
     distance = matDistance (route(i), route(i+1));
     vector_distance = [vector_distance, distance];
     sum_distance = sum(vector_distance);
  end
end

[dpb]

So what's the question/problem?

The total_dist function is adding up the sums of all the preceding distances every time inside the loop may be the issue?

Move it (the sum) outside the loop or just keep a running sum since you aren't returning the vector you're building, anyways, there's no need for it.