I don't get why the #1 odds will decrease each round while the #14 increases.
Because there's a 20% chance we win the 1st lottery, so the odds of the winning the 2nd lottery is the odds we are in the 2nd lottery(80%) multiplied by the odds of winning once we get there(which will be >20 depending on who wins the first lottery)
Once we lose the 1st lottery the odds of winning the 2nd are greater than 20% but before the first lottery happens you have to take into account we might not be in the 2nd lottery if we win the first.
For example lets say 2nd place wins the first lottery. Our odds of winning the 2nd lottery would then be 20/86.5 = 23.12%, the 86.5 coming from all the teams odds for the first lottery minus the 13.5% from the 2nd team that gets taken out of it for the 2nd draw.
So if you just by that number as an estimate for our odds on the 2nd pick once we get there(the lower a team wins it the worse our odds for the 2nd pick would be, for example if the 6th last team won the 1st lottery or odds of the 2nd one is 21.62%, but for simplicity sake lets just use 23.12%).
The our odds of winning the 2nd pick is .80 x 0.2312 = 18.50%, in reality it's actually lower than that because our odds for the 2nd pick get worse if a team lower than 2nd last wins it which is highly likely.
Now after all that to answer you actual question. The 14th last place team is highly unlikely to win the first lottery, they have 1% chance, which means the chance they're in the 2nd lottery is 99%, so when multiplying their odds of winning the 2nd pick by this number it's above 1% so their offs of getting the 2nd pick when all is said and done is higher than their chance of getting the first pick.
You'll notice there's a buffer line in there where teams above it odds of obtaining the 2nd pick is lower than the first pick, and teams below it is higher. Only the top 3 teams have a lesser chance at the 2nd pick than they do at the first pick because their odds of winning the first pick are the higher so the odds they're not in the 2nd draw impacts their overall chance of winning it, which they wouldn't really care about because not being in the 2nd draw means you won the first.
tl;dr: the odds of winning the first pick for teams at the top(or bottom of the standings if you will) is high enough that them winning it and not being in the 2nd draw affects their overall odds of having the 2nd pick, while there's a really high chance the teams with the worse percentages are still in the draw because they didn't win their first pick so their overall odds go up a smidgen.