If you want to say this in reference to people specifically (as required by the use of "aki" in the Hungarian", then use "He who is heavy falls down." If you want to say it more generally (as if the Hungarian sentence used "ami"), then use "That which is heavy falls down." But "That who" at the beginning of the sentence doesn't work.
This sentence structure is common in Hungarian but can seem difficult to translate into English colloquially because English rarely uses the same kind of structure. A useful trick to find a good English equivalent can be to imagine the Hungarian sentence as an answer to a question. The Hungarian sentence will often suggest exactly what question it is answering, because of the word order. For example,
"Az esik le, ami nehéz"
is obviously an answer to the question
"Mi esik le?"
Which is easy to translate into English as
"What falls down?"
and which you can answer simply as
"Heavy things fall down,"
which may be more appropriate in some contexts than the slightly pompous/scientific sounding construction "That which is heavy falls down," which tries to preserve the relative pronoun from Hungarian.
I got this the second time and wondered what the comments would be. With these questions which involve listening and typing, I am interested in the meaning but, primarily, I am typing what I hear. I had to listen to this several times and still found it very difficult to hear what was said. Usually, the speech is quite clear but not in this example. That was my impression.