Can someone explain definition of number 1? I've found the next piece of text:

As an example of the second failing, Poincaré recalled the definition of the number 1 offered by another of the logicists, Burali-Forti:
  $$1 = \imath\,T'\{Ko\cap(u,h)\in(u\in O\,n\,e)\}$$
  This is written in a notation devised by Peano, and indeed in what Poincaré called the "Peanian" language.

And what is the meaning of this equation?
 A: There are many different ways in which the number 'one' can be defined. A discussion of the text you reference can be found in Poincare's "Science in Method" (pp. 458-459) where the author, too, is unsure of Burali-Forti's notation. (Note I am using a translation by Halsted from 1982.)

I understand Peanian [the language of Peano] too ill to dare risk a critique, but still I fear this definition contains a petitio principii, considering that I see the figure 1 in the first number and Un in letters in the second. (Poincare, p. 458)

He discusses a different definition of 'zero', by another mathematician (Couturat), a few lines later:

zero is the number of things satisfying a condition never satisfied. (Poincare, p. 458)

Finally, Poincare cites Couturat again in defining 'one' as follows:

One, says [Couturat] in substance, is the number of elements in a class in which any two elements are identical. (Poincare, p. 458)

Of course, Poincare notes a potentially severe issue in the above definition of one: namely, it relies on the use of the word two!
If you read elsewhere about Peano (including other questions asked here on MSE) I am sure you can see some ways in which 'zero' and 'one' are defined.
I will not try to give a definition here, but remark instead that this section ("Mathematics and Logic") concludes shortly thereafter with a brief exchange between Poincare and Hadamard. The former believes Burali-Forti's reasoning to be "irreproachable" (p. 459) with which Hadamard disagrees, observing further that

Burali-Forti had no right to speak of the aggregate of all the ordinal numbers. (Hadamard cited in Poincare, p. 459)

Poincare then states he tried in vain to convince Hadamard otherwise, but that he was unsuccessful; all for the best, he subsequently remarks, as he [Poincare] feels Hadamard was ultimately correct in the matter.
I include a mention of the short exchange between these two because I find it to be somewhat interesting: This is the sole mention of the mathematician Jacques Hadamard in Poincare's long tome, though Hadamard would go on to write an important piece on creativity in mathematics - one that popularized the classic tale of Poincare's sudden insight about Fuschian functions as he steps onto an omnibus - entitled "Psychology of Invention" (Hadamard, 1945). This (much shorter) book is, according to its author, inspired by Poincare's writing. It doesn't help you with defining natural numbers, but it's another fine "piece of text" that I thought worthy of citation here.
A: The original paper by Cesare Burali-Forti explains this formula as follows:

we say that one is the ordinal type of the ordinal class $(u, h)$ such that $u$ contains only one item.

Here is the Italian original:

Poniamo $$1 = \overline{\imath\,}\,\mathrm{T}`\{\mathrm{Ko}\cap\overline{(u,h)}\mathop{\varepsilon}(u\mathop{\varepsilon}\mathrm{Un})\}$$ cioè diciamo uno il tipo d'ordine delle classi ordinate $(u, h)$ tali che $u$ contiene un solo elemento.

Here the paper uses the older Peano notation $\varepsilon$ for set inclusion that was obsoleted in Russell's Principia. $\textrm{Ko}$ is the class of ordinals, $\mathrm{Un}$ is the class of sets that contain a single element (it can be defined for instance using equality, but this is not discussed in the paper), and $\mathrm{T}`(u,h)$ is the function that maps an equivalence class $(u,h)$ to its ordinal type.
The point of this definition is to link natural numbers and ordinal types, namely stating that natural numbers can be identified with the ordinal types of finite ordinals.
Here is the screenshot of the formula as it appears in the paper called Una Questione sui Numeri Transfiniti that was published in Rendiconti del Circolo Matematico di Palermo in 1897.

