It's a hashing algorithm, helpfully called the HASH algorithm. Cute.

int HASH(string s)
{
    int value = 0;
    for (char c : s)
    {
        value += c;
        value *= 17;
        value %= 256;
    }
    return value;
}

It took longer to read the writeup than to implement that, and sum the input.

Ah, okay. Now it's a HASHMAP. Cute?

Part 2 is significantly more complex... not impossible to deal with, but still complex. I might be able to get away with a map of string arrays, and just swap stuff around as need be?

Either way, just implement as described in part 2:

for (string step : sequence)
{
    if (step.back() == '-')
    {
        string prefix = step.substr(0, step.length() - 1);
        int hash = HASH(prefix);
        
        int removeIndex = -1;
        for (int i = 0; i < boxes[hash].size(); i++)
        {
            if (boxes[hash][i].substr(0, prefix.length()) == prefix)
            {
                removeIndex = i;
                break;
            }
        }

        if (removeIndex >= 0)
        {
            boxes[hash].erase(boxes[hash].begin() + removeIndex);
        }
    }
    else
    {
        auto parts = split(step, "=");
        int hash = HASH(parts[0]);

        int removeIndex = -1;
        for (int i = 0; i < boxes[hash].size(); i++)
        {
            if (boxes[hash][i].substr(0, parts[0].length()) == parts[0])
            {
                removeIndex = i;
                break;
            }
        }

        boxes[hash].push_back(step);
    }
}