I am going to be using multiple number systems so I will explain a few terms to keep this as simple as possible. When referring to a system I will spell out the number (base ten, base eleven). When I am referring to an actual number I will also spell it out and refer to it as a number (the number ten). I will refer to the way a number is represented in a given system as digits and I will use the actual digits to represent it (the digits 10)
First I will explain base ten. The numbers one through ten are represented in base ten as 1-2-3-4-5-6-7-8-9-10. Each digit represents a value and its position determines what it is multiplied by. In 10 the digit 1 is in the second position so it is multiplied by ten.
Now just for comparison I am going to explain base eleven. If we accept X as the digit which represents ten then the following would represent one through ten: 1-2-3-4-5-6-7-8-9-X. In base eleven the position which was multiplied by ten now get multiplied by eleven. The digits 10 represent eleven.
Through these two explanations I hope it is easy for even someone who doesn't fully understand how to use different bases to understand the new system. The base tells how the digits work. The value of 1 in the digit furthest to the right (the first digit) is always one. The value is then multiplied by the base for the next digit. The values of 1 continue to be multiplied by the base for each digit and therefore grow exponentially. To use a digit other than 1 simply multiply the number it represents by the value of 1 in that position.
The idea behind the "New Base Ten" proposal is to use the same number of digits throughout an entire "set" of numbers. An example of this is one through ten. In the new system it would go something like 1-2-3-4-5-6-7-8-9-X. Eleven would still be represented by 11.
The problem with this new system can be seen when one starts working with multiple digit numbers. The first possible two digit number is actually ten in this system. Because it is still base ten, the second digit still multiplies its value by ten. So if the second digit is 1, that represents ten, but still needs a placeholder to show that it is the second digit (the role played by 0 in our current system). This problem can be avoided by using the single digit X, but the problem is even more real when one tries to represent twenty in this system. We don't have a placeholder so we can't use 2 in the second digit. This means in order to represent twenty, one must use the digits 1X. The first digit is X which represents ten, and the second digit is in the place which gets multiplied by the base, ten. This leaves us again with two ways of representing ten.
This is not a very big problem as it is the intended purpose of the new system (to group twenty with eleven through nineteen), but the same problem manifests in a much worse way when working with one-hundred and up. In the new base ten system one hundred should be represented as 9X to group it with eleven through ninety-nine. One-hundred-one should then be represented with a set of three digits (to group it with one-hundred-two through nine-hundred-ninety-nine). Unfortunately, we do not have the placeholder 0 to represent it as 101, so we must settle for X1 (X*ten and 1*one). One-hundred-ten is represented by XX and finally the first three digit number is one-hundred-eleven which is represented by 111. This means that the numbers eleven (represented currently as 11) through one-hundred-ten (represented currently as 110) will all be grouped together as two digit numbers.
This system would simply cause more trouble than it fixes, and it is in everyone's best interest to just accept that ten is actually grouped with one through nine, as is one-hundred with one through ninety-nine.