Macro works only when viewing the source code

Well, if you run it now, you will get an error as today's date can not be negative.
Start = Day(Date - 20) & "." & Month(Date - 20) & "." & Year(Date - 20)
You are also using send keys which are not reliable and do not even use them.
Is the site private, i.e. can you post the URL?

Option Explicit
Sub Download()
Dim y As Long
Dim x As Long
Dim Start As Date
Dim Ending As Date
Dim IE As Object
Dim hTable As Object
Dim hBody As Object
Dim hTR As Object

Dim htD As Object
Dim tb As Object
Dim tr As Object
Dim td As Object

Start = Day(Date - 20) & "." & Month(Date - 20) & "." & Year(Date - 20)
Ending = Day(Date) & "." & Month(Date) & "." & Year(Date)

Set IE = CreateObject("InternetExplorer.Application")
IE.Visible = True
IE.navigate " 'Web site from which I am dowloading data' "

Do Until IE.readyState = 4
Loop

IE.document.all("txtDateFrom").Value = Start
IE.document.all("txtDateTo").Value = Ending
IE.document.all("txtDateTo").Select
SendKeys "{ENTER}", True
SendKeys "{NUMLOCK}", True

Application.Wait (Now + TimeValue("00:00:01"))

Set hTable = IE.document.getElementsByTagName("table")
    For Each tb In hTable
        Set hTR = IE.document.getElementsByTagName("tr")
        For Each tr In hTR
            Set htD = tr.getElementsByTagName("td")
            y = 1
                For Each td In htD
                    Cells(x, y) = td.innerText
                    y = y + 1
                Next td
                    x = x + 1
        Next tr
            Exit For
    Next tb
IE.Quit

End Sub

Okay, this works, but you need to be careful with your dates. They are actually not dates, but strings. The out put data are in sheet1, but you can change it to suit you.

option Explicit
Sub Download()
Dim Start As String
Dim Ending As String
Dim IE As Object
Dim hTable As Object
Dim hBody As Object
Dim hTR As Object
Dim htD As Object
Dim tb As Object
Dim tr As Object
Dim td As Object, c&, r&, t&

Start = Day(Date + 11) & "." & Month(Date - 20) & "." & Year(Date)

Ending = Day(Date) & "." & Month(Date) & "." & Year(Date)

Set IE = CreateObject("InternetExplorer.Application")

    With IE
    'IE.Visible = 0
     .navigate "https://www.conseq.cz/indexes/IndexCZGBFRN.aspx"
    
    Do Until .readyState = 4
    Loop
    
    .document.all("txtDateFrom").Value = Start
    .document.all("txtDateTo").Value = Ending
    .document.all("txtDateTo").Select
End With
SendKeys "{ENTER}", True

SendKeys "{NUMLOCK}", True

Application.Wait (Now + TimeValue("00:00:01"))
Set hTable = IE.document.getElementsByTagName("table")
 For t = 0 To (hTable.Length - 1)

    For r = 0 To (hTable(t).Rows.Length - 1)
        For c = 0 To (hTable(t).Rows(r).Cells.Length - 1)
            Sheets("sheet1").Cells(r + 1, c + 1) = hTable(t).Rows(r).Cells(c).innerText
        Next c
    Next r
 Next t


IE.Quit

End Sub

Thank you for the code, however I have still the same issue, when I call it via button in other Sheet, I still receive only the first row of the table, when I call it from "View code" it works well. Once it downloads the whole table, button works fine too. Could be there an issue in SendKeys method ? Is there a different method to use and "press enter" and send the input on the web page?

Your code errors as you have not given the initial value of x- it is none on the first loop. Initially, I did not notice it. Having correcting it, however, the code still picks up ONLY the first 4 rows. I do not think it is the key issue- more likely to a restriction on the site. I have tried a different method and got the same result as your code. Chrome will not allow you to the see the view code. I suspect the site might have some kind of restriction.

I am using Internet explorer if that matters. I am just wondering, it seems like the Excel has to "see" there is a real user who is actually clicking on View code and then allows to run macro properly. Interesting is when I open the Excel, just click on view code and then Run the macro via button - it works FINE. Seems to me there is maybe some issues in Excel setting rather than website, but honestly I have no clue. Thanks a lot anyway

I am not entirely sure, but I suspect the summation is handled by JavaScript. That is the reason for not seeing any summit button to click. I believe this is done on purpose to avoid multiple session; it is like the kopcha. The default value you get is today's date, even if your dates input are wrong. When you manually directly input the data on the site, you get all the entries. When you do enter the dates through a code, the site is assuming as if your input data are wrong and therefore you get default value , which is a single row of today's date.
I do not know how you are getting all the data through pressing a button. When I run the code using F8, I get only a single row.