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The Elo system with a few modifications (home field advantage, goal difference, inter-league adjustments) is used to determine the club's Elo values. The match odds are based on a result histogram for the two club's Elo difference.
Elo
Elo equation
At the core of these ratings runs the Elo system, a widely used method to measure relative strength levels.
The advantage of Elo lies in its simplicity, there is only one value per club for each point in time, the higher the better.
The Elo difference between two clubs is directly linked to the win probability of both teams in a direct confrontation (draws counting as half win/ half loss):
E = 1 / (10(-dr/400) + 1),
where dr is the Elo point difference of the 2 clubs.
This is the Elo equation.

Points Exchange
When clubs play each other and win or lose, they exchange points. The number of points exchanged must be determined so that a certain win rate between two clubs makes the Elo difference between both clubs converge towards the Elo difference that corresponds to this win rate.
The following equation satisfies this constraint:
ΔElo_1X2 = (R - E) * k,
where R is the result (1 for win, 0.5 for draw, 0 for loss).
There is one degree of freedom in this equation which in the weight index k that has to be chosen. A higher k will have the ratings converge quicker to their true values but will suffer from more variation. A smaller k provides more stable values that take longer to converge.
ClubElo uses a weight index of k = 20.

Tilt
ClubElo introduces a second club metric next to Elo which is named Tilt.
Tilt is designed to be a measure of offensiveness (not quality, that is what Elo is for).
More specifically, it measures if there are more or less total goals than expected in games involving the club. The expectation refers to the long term average of games at the same Elo difference.
Every game has an expected number of total goals. After a game, the tilt of a team (which is initialised at 1) is updated in the following way:
New_tilt = 0.98 * Old_tilt + 0.02 *Game_total_goals/Opposition_tilt/Exp_Game_total_goals
This adds another dimension to predicting exact results and can be shown to add accuracy. It is sometimes expressed as percentage.

Weighting goal difference
A win by a high margin is considered to be more significant than a win by a narrow margin. This is taken into account when the number of exchanged points is calculated. The points exchanged increase proportionally to the square root of the margin. At the same time, the expected number of points exchanged for a win or a loss should be according to the above Elo equation.
The points exchanged for a win or a loss are determined through the following equation:
ΔElo_margin = ΔElo_1goal * sqrt(margin), where
ΔElo_1goal = ΔElo_1X2 / sum(sqrt(margin)*p_margin/p_1X2)
p_margin is the likelyhood for a specific margin, p_1X2 is the likelyhood to win (or lose) by any margin. The sum is over all margins for a win or all margins for a loss.

Home Field Advantage
Clubs have from experience a higher probability to win games when they play at home. This would mean that on average home teams would win more Elo points than away teams.
This would violate the constraint that the expected number of Elo points exchanged per match should always be 0. That is why ClubElo increases the Elo difference for a match by a certain number of Elo points, called HFA (for Home Field Advantage).
As HFA can initially only be guessed, an algorithm is in place to make sure that this value converges to where it should be:
Every day and for every country seperately, the system compares if home teams won more or less points than away teams. If home teams won more, HFA is increased, if away teams won more points, HFA is decreased according the the following equation:
HFA += sum(ΔElo)*0.075

Two-leg matches
Two-leg matches are considered one long match. The aggregated result over two legs determines the total number of exchanged points (sum of points exchanged after leg 1 and 2). Compared to a single game, the exchanged points are therefore multiplied by the square root of 2. A victory by away goals rule counts as a victory by a margin of half a goal.
The points exchanged after the first leg are determined by the difference in outcome likelyhoods before leg one and after leg 1.