Tischendorf's own account of what happened: http://rosetta.reltech.org/TC/extras/tischendorf-sinaiticus.html
"It was in April, 1844, that I embarked at Leghorn for
Egypt. The desire which I felt to discover some precious remains of any manuscripts, more especially Biblical, of a date which would carry us back to the early times of Christianity, was realized beyond my expectations. It was at the foot of
Mount Sinai, in
the Convent of St. Catherine, that I discovered the pearl of all my researches. In visiting the library of the monastery, in the month of
May, 1844, I perceived in the middle of the great hall a large and
wide basket full of old parchments; and the librarian, who was a man of information, told me that
two heaps of papers like these, mouldered by time, had been already committed to the flames. What was my surprise to find amid this heap of papers a considerable number of sheets of a copy of the Old Testament in Greek, which seemed to me to be one of the most ancient that I had ever seen.
The authorities of the convent allowed me to possess myself of a third of these parchments, or about forty-three sheets, all the more readily as they were destined for the fire." (Tischendorf)
Here Tischendorf explains how he found the manuscript in the waste-basket that was containing manuscripts to be burned to heat the room. He is only allowed to 43 sheets.
"...and enjoined on the monks to take religious care of all such remains which might fall in their way."
In other words, he said "I'm not content with these 43 sheets but I want the whole manuscript, so you take it out of the trash and hold on to it and I'll be back with money."
"I resolved, therefore, to return to the East to copy this priceless manuscript. Having set out from Leipzig in January, 1853, I embarked at Trieste for Egypt, and in the month of February I stood for the second time in the Convent of Sinai." (Tischendorf)
"But I felt myself more and more urged to recommence my researches in the East...for the Emperor of Russia, led me, in the autumn of 1856, to submit to the Russian Government a plan of a journey for making systematic researches in the East...People were astonished that a foreigner and a Protestant should presume to ask the support of the Emperor of the Greek and Orthodox Church for a mission to the East...It obtained his approval in the month of September, 1858,
and the funds which I asked for were placed at my disposal. Three months subsequently my seventh edition of the New Testament, which had cost me three years of incessant labour, appeared; and in the commencement of January, 1859, I again set sail for the East." (Tischendorf)
***I cannot here refrain from mentioning the peculiar satisfaction I had experienced a little before this. A learned Englishman, one of my friends, had been sent into the East by his Government to discover and purchase old Greek manuscripts, and spared no cost in obtaining them. I had cause to fear, especially for my pearl of the Convent of St. Catherine; but I heard that he had not succeeded in acquiring anything, and had not even gone as far as Sinai--"for," as he said in his official report, "after the visit of such an antiquarian and critic as Dr. Tischendorf, I could not expect any success." I saw by this how well advised I had been to reveal to no one my secret of 1844.*** (Tischendorf)
"...he [monk at St. Catherines] took down from the corner of the room a bulky kind of volume, wrapped up in a red cloth, and laid it before me. I unrolled the cover, and discovered, to my great surprise, not only those very fragments which, fifteen years before, I had taken out of the basket, but also other parts of the Old Testament, the New Testament complete, and, in addition, the Epistle of Barnabas and a part of the Pastor of Hermas." (Tischendorf)
What's he saying here is that the monk from 15 years ago kept the manuscript he found, waiting for the money that had been promised so long ago. What the monk had wrap in the cloth was what he took out of the trash 15 years ago in anticipation of receiving a large sum of money from Tischendorf in the future, as Tischendorf himself explains.